Editing Proportional–integral–derivative controller (section)

===Discrete implementation===
The analysis for designing a digital implementation of a PID controller in a [[microcontroller]] (MCU) or [[FPGA]] device requires the standard form of the PID controller to be ''discretized''.<ref>{{cite web |url=https://www.scribd.com/doc/19070283/Discrete-PI-and-PID-Controller-Design-and-Analysis-for-Digital-Implementation |title=Discrete PI and PID Controller Design and Analysis for Digital Implementation |publisher=Scribd.com |access-date=2011-04-04 |url-access=registration}}</ref> Approximations for first-order derivatives are made by backward [[finite difference]]s. <math>u(t)</math> and <math>e(t)</math> are discretized with a sampling period <math>\Delta t</math>, k is the sample index.

Differentiating both sides of PID equation using [[Newton's notation]] gives:

<math>\dot{u}(t) = K_p\dot{e}(t) + K_ie(t) + K_d\ddot{e}(t)</math>

Derivative terms are approximated as,
:<math>\dot{f}(t_k) = \dfrac{df(t_k)}{dt}=\dfrac{f(t_{k})-f(t_{k-1})}{\Delta t}</math>

So,

:<math>\frac{u(t_{k})-u(t_{k-1})}{\Delta t} = K_p\frac{e(t_{k})-e(t_{k-1})}{\Delta t} + K_i e(t_{k}) + K_d \frac{\dot{e}(t_{k}) - \dot{e}(t_{k-1})}{\Delta t}</math>

Applying backward difference again gives,

:<math>\frac{u(t_{k})-u(t_{k-1})}{\Delta t} = K_p\frac{e(t_{k})-e(t_{k-1})}{\Delta t} + K_i e(t_{k}) + K_d \frac{ \frac{e(t_{k})-e(t_{k-1})}{\Delta t} -  \frac{e(t_{k-1})-e(t_{k-2})}{\Delta t} }{\Delta t}</math>

By simplifying and regrouping terms of the above equation, an algorithm for an implementation of the discretized PID controller in a MCU is finally obtained:

:<math>u(t_{k})=u(t_{k-1})+\left(K_p+K_i\Delta t+\dfrac{K_d}{\Delta t}\right) e(t_{k})+\left(-K_p-\dfrac{2K_d}{\Delta t}\right) e(t_{k-1}) + \dfrac{K_d}{\Delta t}e(t_{k-2})</math>

or:

:<math>u(t_k)=u(t_{k-1})+K_p\left[\left(1+\dfrac{\Delta t}{T_i}+\dfrac{T_d}{\Delta t}\right) e(t_k)+\left(-1-\dfrac{2T_d}{\Delta t}\right)e(t_{k-1}) + \dfrac{T_d}{\Delta t}e(t_{k-2})\right]</math>

s.t. <math> T_i = K_p/K_i, T_d = K_d/K_p</math>

Note: This method solves in fact <math>u(t) = K_\text{p} e(t) + K_\text{i} \int_0^t e(\tau) \,\mathrm{d}\tau + K_\text{d} \frac{\mathrm{d}e(t)}{\mathrm{d}t} + u_0</math> where <math>u_0</math> is a constant independent of t. This constant is useful when you want to have a start and stop control on the regulation loop. For instance, setting Kp,Ki and Kd to 0 will keep u(t) constant. Likewise, when you want to start a regulation on a system where the error is already close to 0 with u(t) non null, it prevents from sending the output to 0.